where \(\mathcal {O}_p\)  is the Orthogonal manifold, that is the set of matrices \(X \in \mathbb {R}^{p\times p}\)  such that \(XX^{\top } = I_p\) .
Problem (REF ) appears in many practical applications, like principal component analysis, independent component analysis [1]}, [2]}, [3]}, procrustes problem [4]}, and more recently in deep learning, where the  weights of a layer are parametrized by an orthogonal matrix [5]}, [6]}.
This is a particular instance of minimization over a matrix Riemannian manifold, \(\mathcal {O}_p\)  [7]}.
Many standard Euclidean algorithms for function minimization have been adapted on Riemannian manifolds.
We can cite for instance gradient descent [8]}, [9]}, second order quasi-Newton methods [10]}, [11]}, and stochastic methods [12]} which are the workhorse for training deep neural networks.
More recently, several works propose to adapt accelerated methods in the Riemannian setting [13]}, [14]}.
<FIGURE>